Trigonometric Ratios:
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 1
Sin Ɵ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
; Cosec Ɵ = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
; Cosec Ɵ = 𝑆𝑖𝑛 Ɵ
;
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 1
Cos Ɵ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
; Sec Ɵ = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
; Sec Ɵ = 𝐶𝑜𝑠 Ɵ
;
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 1
Tan Ɵ = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
; Cot Ɵ = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
; Cot Ɵ = 𝑡𝑎𝑛 Ɵ
;
Sin, Cos, Tan, Cosec, Sec, Cot are short names for sine, cosine, tangent, cosecant, secant and
cotangent respectively.
Trigonometric Ratios of Special Angles:
First triangle is an isosceles right angled triangle whereas the second triangle is an equilateral
triangle. Properties of the triangle along-with Pythagoras theorem can be used to find
trigonometric ratios of angles 45˚, 30˚, 60˚. A complete table with values is below.
Angle Ɵ 0˚ or 0 30˚ or
π
45˚ or
π
60˚ or
π
90˚ or
π 180˚ or π 270˚ or
6 4 3 2 3π
2
Sin Ɵ 0 1 1 √3 1 0 -1
2 √2 2
Cos Ɵ 1 √3 1 1 0 -1 0
2 √2 2
Tan Ɵ 0 1 1 √3 ∞ 0 -∞
√3
Area of Triangle:
1
(a) Area = 2
x base x height
(b)When two sides and included angle is given,
1
Area = 2
abSinC
(c) Equilateral Triangle
√3
Area = 4
a²
(d)Heron’s Formula
𝑎+𝑏+𝑐
S= 2
Area = 𝑆(𝑆 − 𝑎)(𝑆 − 𝑏)(𝑆 − 𝑐) where S is semi-perimeter of the triangle and a, b, c
are the lengths of the sides.
Trigonometric Identities:
(a) Basic Identities: Sin²x + Cos²x = 1
1 + Tan²x = Sec²x
1 + Cot²x = Cosec²x
(b)Complementary Angle Identities:
Sin (90˚-Ɵ) = Cos Ɵ
Cos (90˚-Ɵ) = Sin Ɵ
Tan (90˚-Ɵ) = Cot Ɵ
Cosec (90˚-Ɵ) = Sec Ɵ
Sec (90˚-Ɵ) = Cosec Ɵ
Cot (90˚-Ɵ) = Tan Ɵ
(c) Trigonometric Ratios of Sum & Difference of Angles:
Sin (A+B) = Sin A Cos B + Cos A Sin B
Sin (A-B) = Sin A Cos B – Cos A Sin B
Cos (A+B) = Cos A Cos B – Sin A Sin B
Cos (A-B) = Cos A Cos B + Sin A Sin B
𝑇𝑎𝑛 𝐴+𝑇𝑎𝑛 𝐵
Tan (A+B) = 1−𝑇𝑎𝑛 𝐴 𝑇𝑎𝑛 𝐵
𝑇𝑎𝑛 𝐴−𝑇𝑎𝑛 𝐵
Tan (A-B) = 1+𝑇𝑎𝑛 𝐴 𝑇𝑎𝑛 𝐵
(d)Double Angle Formulas: Sin 2x = 2Sin x Cos x
2𝑇𝑎𝑛𝑥
= 1+𝑇𝑎𝑛²𝑥
Cos 2x = Cos²x - Sin²x
= 2Cos²x – 1
= 1 – 2Sin²x
1−𝑇𝑎𝑛²𝑥
= 1+𝑇𝑎𝑛²𝑥
2𝑇𝑎𝑛𝑥
Tan 2x = 1−𝑇𝑎𝑛²𝑥
𝑥 1+𝐶𝑜𝑠𝑥
(e) Half-Angle Formulas: Cos 2
= 2
𝑥 1−𝐶𝑜𝑠𝑥
Sin 2
= 2
𝑥 1−𝐶𝑜𝑠𝑥
Tan 2
= 1+𝐶𝑜𝑠𝑥
Cosine Rule:
a² = b² + c² - 2bcCos A b² = c² + a² - 2caCos B c² = a² + b² -
2abCos C
or or or
2 2 2 2 2 2
𝑏 +𝑐 −𝑎² 𝑐 +𝑎 −𝑏² 𝑎 +𝑏 −𝑐²
Cos A = 2𝑏𝑐
Cos B = 2𝑐𝑎
Cos C = 2𝑎𝑏
Sine Rule:
Ambiguous Case of sine rule: For SSA case, Sine rule gives two possible angles.
It’s clear from above construction that two triangles are possible with length of two sides 9 and 7
and one non-included angle being 45˚.
Using Sine Rule, we can find angle C.
=> Sin C = 0.9091
=> C = 65.4˚ or 114.6˚
Clearly ∠ C has two possible values. As unique triangle is not possible, this is an example of
ambiguous case.
Radians: 180˚ = π rad
Area of Sectors: Area of Sector = πr²x = , if Ɵ is in radians.
= x , if Ɵ is in degrees.
Length of Arc of a circle: l = rƟ, if Ɵ is in radians.
Ɵ
= 2πr x 360˚
, if Ɵ is in degrees.
Circular Trigonometric Functions:
Unit circle helps us in redefining trigonometric functions for angles greater than 90˚. From the
diagram above, Cos Ɵ & Sin Ɵ both are positive as they lie in 1st quadrant. If angle Ɵ lies in
some other quadrant, signs of Cos Ɵ and Sin Ɵ may change depending on the quadrant. Clearly,
Cos Ɵ follows sign of x co-ordinate and Sin Ɵ follows sign of y co-ordinate.
Thus, if Ɵ is the angle between OP and the positive x-axis:
• the cosine of Ɵ is defined to be the x-coordinate of the point P on the unit circle
• the sine of Ɵ is defined to be the y-coordinate of the point P on the unit circle.
From the above diagram,
Cos 150˚ = - Cos 30˚ as x co-ordinate is –ve in 2nd
Sin 150˚ = Sin 30˚ as y co-ordinate is +ve in 2nd
CAST Diagram:
Sin is +ve All are +ve S A
is abbreviated as
Tan is +ve Cos is +ve T C
Reference Angle & Trigonometric Ratio Formulas for Angles in 2nd, 3rd & 4th quadrant:
When the rotation angle is greater than 90˚, the acute angle between the rotation line and the
x-axis is the reference angle.
2nd Quadrant 1st Quadrant
180˚-Ɵ or
π–Ɵ
π + Ɵ or 2π – Ɵ or
180˚ + Ɵ 360˚ - Ɵ
3rd quadrant 4th Quadrant
(a) 2nd Quadrant Formulas: Sin (180˚ - Ɵ) = Sin Ɵ or Sin (π - Ɵ) = Sin Ɵ
Cos (180˚ - Ɵ) = - Cos Ɵ or Cos (π - Ɵ) = - Cos Ɵ
Tan (180˚ - Ɵ) = - Tan Ɵ or Tan (π - Ɵ) = - Tan Ɵ
(b)3rd Quadrant Formulas: Sin (180˚ + Ɵ) = - Sin Ɵ or Sin (π + Ɵ) = - Sin Ɵ
Cos (180˚ + Ɵ) = - Cos Ɵ or Cos (π + Ɵ) = - Cos Ɵ
Tan (180˚ + Ɵ) = Tan Ɵ or Tan (π + Ɵ) = Tan Ɵ
(c) 4th Quadrant Formulas: Sin (360˚ - Ɵ) = - Sin Ɵ or Sin (2π - Ɵ) = - Sin Ɵ
Cos (360˚ - Ɵ) = Cos Ɵ or Cos (2π - Ɵ) = Cos Ɵ
Tan (360˚ - Ɵ) = - Tan Ɵ or Tan (2π - Ɵ) = - Tan Ɵ
Trigonometric Equations:
(a) Cos Ɵ = Cos α, where α is a constant angle.
=> Ɵ = 2nπ ± α
(b) Sin Ɵ = Sin α, where α is a constant angle.
=> Ɵ = nπ + (-1)ⁿα
(c) Tan Ɵ = Tan α, where α is a constant angle.
=> Ɵ = nπ + α
Example 1: 2Sin²t = 1
1 1 π 7π
=> Sin²t = 2
=> Sin t = ± √2 => t = nπ + (-1)ⁿ 4 , nπ + (-1)ⁿ 4
π 3π 5π 7π
Solutions between 0 and 2π are 4
, 4
, 4
, 4
.
Example 2: Cos 2Ɵ + Cos Ɵ + 1 = 0
=> 2Cos²Ɵ – 1 + Cos Ɵ + 1 = 0
=> 2Cos²Ɵ + Cos Ɵ = 0
=> Cos Ɵ ( 2 Cos Ɵ + 1 ) = 0
−1
=> Cos Ɵ = 0, 2
.
π 2π
=> Ɵ = 2nπ ± 2
, 2nπ± 3
.
π 2π 4π 3π
Solutions between 0 and 2π are 2
, 3
, 3
, 2
.
Inverse Trigonometric Functions:
Sin Ɵ = x
=> Ɵ = Sin⁻¹x
Usually Sin⁻¹x can be equal to any angle whose Sine is x. So, it’s a multi-valued function.
Whereas arcsinx is a one-to-one function and usually represents principal values.
Domain Range
π π
arcsinx [-1,1] [- 2 , 2
]
arccosx [-1,1] [0, π]
π π
arctanx (-∞, ∞) (- 2 , 2
)
Example 1: tan(arctan2) = 2.
5π −√3 2π
Example 2: arcsin(sin 3
) = arcsin( 2
)=- 3
.
2
Example 3: tan(arccos 3
)=?
2
Let arccos 3
=x
2
=> Cos x = 3
2
3 −2² √5
=> Tan x = 2
= 2
2 √5
=> tan(arccos 3
)= 2
Ans.
Trigonometric Graphs:
(a) Sine function
Domain = R
Range = [-1, 1]
Period = 2π
(b)Cosine function
Domain = R
Range = [-1, 1]
Period = 2π
(c) Tangent Function
π
Domain is set of all real numbers except odd multiples of 2
.
Range = R
Period = π
Asymptotes
π
x = (2n+1) 2
(d)Sinusoidal Graphs
y = A Cos [ B ( x + C ) ] + D
Here, graph of Cos x has been transformed with stretches and translations.
C Period
|A|
Axis
D
|A| is the height of the curve above the axis. It’s also called amplitude.
C is translation left or right parallel to the x-axis ( negative when it’s to the right and
positive when it’s to the left )
D is translation up or down parallel to the y-axis ( negative when it’s down or
positive when it’s up )
360˚ 2π
B is stretch or squash parallel to the x-axis. B = 𝐵
= 𝐵
.
